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Solution of the differential equation 3y dy/dx + 2x = 0 represents a family of?
- a)Ellipses
- b)Circles
- c)Parabolas
- d)Hyperbolas
Correct answer is option 'A'. Can you explain this answer?
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Solution of the differential equation 3y dy/dx + 2x = 0 represents a ...
3y dy/dx + 2x = 0
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⇒ ∫ 3y dy = ∫ 2x dx
⇒ 3y2/2 = -2x2/2 + c ( on integrating )
⇒ x2 + 3y2/2 = c or x2/1 + y2/2/3 = c
x2/c + y2/2c/3 =1
which represents a family of ellipse.
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Solution of the differential equation 3y dy/dx + 2x = 0 represents a ...
Solution:
The given differential equation is 3y(dy/dx) + 2x = 0.
To solve this differential equation, we can rearrange the terms as follows:
3y(dy/dx) = -2x
Now, we can separate the variables by bringing all the y terms to one side and all the x terms to the other side:
dy/y = (-2/3)x dx
Integrating both sides, we get:
∫(dy/y) = ∫(-2/3)x dx
ln|y| = (-1/3)x^2 + C
where C is the constant of integration.
Now, we can exponentiate both sides to eliminate the natural logarithm:
|y| = e^((-1/3)x^2 + C)
Simplifying further, we get:
|y| = e^C * e^(-1/3)x^2
Since e^C is just another constant, let's denote it as A:
|y| = A * e^(-1/3)x^2
Now, we can consider two cases: when y is positive and when y is negative.
Case 1: y > 0
In this case, |y| = y, so we have:
y = A * e^(-1/3)x^2
Case 2: y < />
In this case, |y| = -y, so we have:
-y = A * e^(-1/3)x^2
Multiplying both sides by -1, we get:
y = -A * e^(-1/3)x^2
Therefore, the general solution of the given differential equation is:
y = ±A * e^(-1/3)x^2
This represents a family of curves, where each curve is a parabola.
Hence, the correct answer is option C) Parabolas.
The given differential equation is 3y(dy/dx) + 2x = 0.
To solve this differential equation, we can rearrange the terms as follows:
3y(dy/dx) = -2x
Now, we can separate the variables by bringing all the y terms to one side and all the x terms to the other side:
dy/y = (-2/3)x dx
Integrating both sides, we get:
∫(dy/y) = ∫(-2/3)x dx
ln|y| = (-1/3)x^2 + C
where C is the constant of integration.
Now, we can exponentiate both sides to eliminate the natural logarithm:
|y| = e^((-1/3)x^2 + C)
Simplifying further, we get:
|y| = e^C * e^(-1/3)x^2
Since e^C is just another constant, let's denote it as A:
|y| = A * e^(-1/3)x^2
Now, we can consider two cases: when y is positive and when y is negative.
Case 1: y > 0
In this case, |y| = y, so we have:
y = A * e^(-1/3)x^2
Case 2: y < />
In this case, |y| = -y, so we have:
-y = A * e^(-1/3)x^2
Multiplying both sides by -1, we get:
y = -A * e^(-1/3)x^2
Therefore, the general solution of the given differential equation is:
y = ±A * e^(-1/3)x^2
This represents a family of curves, where each curve is a parabola.
Hence, the correct answer is option C) Parabolas.
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Solution of the differential equation 3y dy/dx + 2x = 0 represents a family of?a)Ellipsesb)Circlesc)Parabolasd)HyperbolasCorrect answer is option 'A'. Can you explain this answer?
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